Lesson 4

Scaling and Area

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Problem 1

A rectangle with area 12 square units is dilated by a scale factor of \(k\). Find the area of the image for each given value of \(k\).

  1. \(k=2\)
  2. \(k=5\)
  3. \(k=1\)
  4. \(k=\frac14\)
  5. \(k=1.2\)

Solution

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Problem 2

The area of a circle of radius 1 is \(\pi\) units squared. Use scaling to explain why the area of a circle of radius \(r\) is \(\pi r^2\) units squared.

Solution

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Problem 3

Trapezoid \(A’B’C’D\) was created by dilating trapezoid \(ABCD\) using \(D\) as the center of dilation.

A large trapezoid A'B'C'D' comprised of smaller shapes on a grid.
  1. What was the scale factor of the dilation?
  2. Based on the scale factor, how many copies of \(ABCD\), including the original, should fit inside \(A’B’C’D\)?
  3. How can you see your answer to these questions in the diagram?

Solution

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Problem 4

Each image shows a quadrilateral in a plane. The quadrilateral has been dilated using a center above the plane and a scale factor between 0 and 1. Estimate the scale factor that was used for each dilation.

Dilation A

A quadrilateral in a plane.

Dilation B

A quadrilateral in a plane.

Dilation C

A quadrilateral in a plane.

Solution

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(From Unit 5, Lesson 3.)

Problem 5

Select the solid whose cross sections are dilations of some two-dimensional shape using a point directly above the shape as a center and scale factors ranging from 0 to 1.

A:

cone

B:

cube

C:

cylinder

D:

triangular prism

Solution

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(From Unit 5, Lesson 3.)

Problem 6

Select all figures for which at least one cross section is a circle.

A:

triangular pyramid

B:

square pyramid

C:

rectangular prism

D:

cube

E:

cone

F:

cylinder

G:

sphere

Solution

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(From Unit 5, Lesson 2.)

Problem 7

If the two-dimensional figures are rotated around the vertical axes of rotation shown, what solids are formed?

A vertical dashed line with a part of a rectangle on the right side of the line.
Vertical dashed line with a part of a circle on the right side of the line.

Solution

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(From Unit 5, Lesson 1.)

Problem 8

Tyler and Jada wish to find the value of \(x\), the length of side \(BC\) in this triangle. Tyler decides to set up the equation \(\tan(56)=\frac8x\). Jada says she prefers an equation that has \(x\) in the numerator. What is an equation she could use instead?

Triangle A B C with angle C marked as a right angle angle B is 56 degrees, A C is 8 units, and B C is x units.

​​​​

Solution

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(From Unit 4, Lesson 8.)

Problem 9

Triangles \(ACD\) and \(BCD\) are isosceles. Angle \(DBC\) has a measure of 110 degrees and angle \(BDA\) has a measure of 22 degrees. Find the measure of angle \(BAC\).

\(\overline{AD} \cong \overline{AC}\) and \(\overline{BD} \cong \overline{BC}\)

Triangle A C D. Point B lies inside A C D. Segments B A, B C, and B D are drawn in, creating 3 smaller triangles. Sides D A and A C have single tick marks. Sides D B and B C have double tick marks.

Solution

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(From Unit 2, Lesson 6.)